On the growth of linear recurrences in function fields

Let $(G_n)_{n=0}^{\infty}$ be a non-degenerate linear recurrence sequence with power sum representation $G_n = a_1(n) \alpha_1^n + \cdots + a_t(n) \alpha_t^n$. In this paper we will prove a function field analogue of the well known result that in the number field case, under some non-restrictive conditions, for $n$ large enough the inequality $\vert G_n\vert \geq \left( \max_{j=1,\ldots,t} \vert \alpha_j\vert \right)^{n(1-\varepsilon)}$ holds true.